Can Financial Assets Beat Social Security? Not in the Real World.

a report by
John Mueller
Senior Vice President & Chief Economist
Lehrman Bell Mueller Cannon, Inc.for the National Committee to Preserve Social Security and Medicare
Washington, D.C.
October 1997

1. An Overview of the Debate

Since retirees began collecting Social Security benefits in 1941, the average real return on payroll taxes paid has been about 9% — far above the average returns in the stock market.[i]

Moreover, until the late 1970s, most economists believed that, while future returns could not remain so high, the average long-run return on pay-as-you-go[ii] Social Security would approximate the rate of economic growth; and that this rate of return must exceed the average return on financial investments of comparable risk.[iii]

About 25 years ago, economists like Martin Feldstein began to question this conclusion.[iv] Feldstein agreed that the long-term return on Social Security would equal the rate of economic growth. But the return on Social Security, according to Feldstein, must be compared, not with a low-risk investment like Treasury bills, but with the total pretax return on business investment.

“The average growth of real wages since 1960 — 2.6 percent — can serve as a reasonable estimate of what an unfunded Social Security program can yield over the long-term future. In contrast, the the real pretax return on nonfinancial corporate capital (i.e., profits before all taxes plus the net interest paid) averaged 9.3 percent over the same period. Although individuals do not earn the full 9.3 percent return even in Individual Retirement Accounts (IRAs) and 401(k) accounts because of federal and state corporate taxes, a funded retirement system could deliver the full 9.3 percent pretax return to each individual saver if the government credited back the corporate tax collections.”[v]

Feldstein and others seek to “privatize” Social Security — replace it with government-mandated financial savings accounts.[vi] But most “privatizers” do not go as far as Feldstein — proposing that all Federal, state and local taxes on the investment financed by retirement saving be abolished. Instead, they argue that, even without major changes in taxation, the future average return on financial assets like stocks and bonds will exceed the return on pay-as-you-go Social Security.

For example, they point out, the average annual real return on common stocks since 1926 has been about 7.4%, while real economic growth averaged about 3.2%.

Usually, “privatizers” push their case still further, comparing past average real returns on the stock market with hypothetical future real returns on Social Security as low as 1% — or even past hypothetical returns of 9% on investment in plant and equipment with future hypothetical returns on Social Security.

To deal with one fallacy at a time, we examine different aspects of the “privatizers'” argument in a series of different papers.

In the current paper, we compare past returns on financial assets with the rate of real economic growth — which all agree is the average long-run real return on pay-as-you-go Social Security. We show that, adjusted for the difference in risk, the return on financial assets has been much lower than the rate of economic growth — and thus below the long-term rate of return on Social Security under the same economic conditions.

In a second paper, we show how to determine the future rate of return on the stock market implied by projections of slower economic growth and changing demographic trends. Under the assumptions of the Social Security actuaries, the average real return on the stock market will be about 1.4% over the next 20 years, and about 3.2% over the next 75 years. The risk-adjusted return on financial assets would remain significantly below the return on Social Security.

In a third paper, we examine the economic theory of Social Security and show what’s wrong with Feldstein’s reasoning as cited above: he ignores the existence of “human capital.” Feldstein compares the pretax return on the investment in nonhuman capital financed by retirement saving placed in stocks and bonds, with the after-tax return on the investment in “human capital” financed by pay-as-you Social Security. The pretax rate of return on human capital is significantly higher than on nonhuman capital. This reinforces the conclusion that ending Social Security would lower, not raise, the return on retirement saving.

2. The “Privatizers'” Main Argument — and Its Flaw.

The main argument of those who seek to end pay-as-you-go Social Security has always been quite simple: “While those retired today are still receiving above-market returns through social security, those now entering the work force are offered low, below-market returns.”[vii]

The argument is typically reinforced by comparing the average real return on the stock market since the end of 1925 (about 7%), or on a mix of stocks and bonds (4% to 5%), with the average real rate of economic growth (about 3% in the past, but possibly lower in the future). The real return on Social Security in the long run should equal the rate of economic growth.

The particular numbers chosen for comparison differ. But the key point, according to the “privatizers,” is that future rates of return on financial investments like the stock market will be higher than the rate of economic growth: “Indeed, any private returns higher than the social security returns discussed earlier would result in higher benefits through the private investment system than through social security.”[viii]

Thus, the “privatizers” conclude, future generations would be better off if pay-as-you-go retirement benefits were replaced by financial savings accounts invested in stock and bonds.

The argument is simple and clear. It is also remarkable for its basic assumption: that investors are indifferent to risk. What the “privatizers” overlook is that, while the return on Social Security is tied to the growth of the economy, so is its volatility– which is only a quarter of the risk of the stock market. And for nearly all investors, the extra return on the stock market is not large enough to offset its extra risk.

If risk could be ignored, no one would (or should) invest in the stock market. This is because many investments promise a much higher return. For example, the three-month Treasury bills of the Turkish government recently have yielded more than 100% — a compound annual return of more than 250%! Since the “privatizers” talk about the U.S. stock market, and not Turkish bonds, it implies that even they recognize the reality of risk — though they leave it out of their calculations.

But investors wouldn’t need to go abroad to raise average returns — if risk didn’t matter. Virtually any average real yield could be manufactured, simply by borrowing enough and investing in almost any asset. The average 7% real yield on common stocks could be doubled to 14%, merely by buying the stocks with 50% “margin” — borrowing an amount equal to the investor’s wealth, and buying twice as many stocks. Borrowing twice as much would yield an average of 21%, borrowing three times as much 28%, and so on.

The reason that no sane person does this on any scale is that the risk is multiplied by the same proportion. Any sensible person would obviously prefer a sure return of 9% or 7% to an equally sure return of 3%. But only a compulsive gambler actually chooses a portfolio with an average real return of 9% or 7%, over one with an average return of 3%, or even 1% — if each entails the average risk actually associated with such average returns in the past 70 years.

3. Measuring Risk and Risk Aversion: A Dollar in Hand. . .

To compare apples with apples, we need to adjust the returns on different investments for their different risks.

What is risk? In a technical sense, risk is simply the probability of a loss. But in investment, risk has two key aspects.

The first is that almost everyone prefers to avoid risk. This means that investors require a financial reward for undertaking risk, and this “risk premium” rises faster than risk.

The second aspect is that a higher average return on investment generally requires accepting higher risk — and over the range of possible investments, risk rises faster than return.

These two factors combined explain why investors as a group do not seek the highest possible average return, but rather the highest risk-adjusted return.

Though most of us don’t use the term “risk aversion,” we all know what it means. The idea is captured in the adage, “A bird in hand is worth two in the bush.” To be risk-averse means that the prospect of losing a dollar you already own weighs more heavily than the chance of gaining a dollar you don’t yet own.[ix]

You can easily find out whether you are risk-averse. We agree to flip a coin. If the coin comes up tails, you lose half your wealth — half your bank accounts, stocks, bonds, house, car and other assets; and also half of what you will earn for the rest of your life. If the coin comes up heads, you win an equal amount.

A “risk-neutral” investor — one who neither seeks nor avoids risk — would just accept this bet, because it is “actuarially fair”: the odds of winning or losing are equal, and so are the potential gains and losses.

If you would not accept the bet, you are risk averse. And you are not alone. Risk aversion is the rational response to the human condition: none of us lives long enough or has enough resources to try risky things an infinite number of times.

Moreover, we have just settled the question whether you are the sort of person who would be better or worse off if pay-as-you-go Social Security were abolished. Those who would be better off are the few who regard a bird in the bush as equal to a bird in hand. You, on the other hand, would be worse off; how much worse depends on how much you prefer to avoid risk.

Almost no one would risk half his wealth on a coin toss. (If you would accept the bet, please see me about some investment ideas that could prove to be mutually advantageous.) But there are degrees of aversion to risk. Most people would accept the bet if it were modified — so that the risk of loss were smaller, the promised payoff larger, or the odds of winning better. It’s possible to measure your risk aversion by comparing how large a gain, and at what odds, would induce you to risk losing some specified share of your wealth.

Economists do a lot of theorizing about risk aversion, but seldom pay much attention to its measurement.[x] The evidence indicates that, for the typical investor, a bird (or a dollar) in hand is indeed worth two in the bush (or in the stock market) — at least, if the number of birds or dollars at stake is not too large compared with the number you start with.

For example, to balance the chance of losing 1% of his wealth, the typical investor requires an equal chance of gaining about 2%. A more conservative investor requires an equal chance of gaining about 3%. A more speculative investor might require only a 1-1/2% gain. (An investor indifferent to risk, of course, would risk a 1% loss for a gain of only 1%.)

What this means, mathematically, is that the value a typical investor places on each extra dollar of wealth varies inversely with the square of his wealth.[xi] For the conservative investor, this “marginal utility” of wealth varies inversely with the cube of his wealth. As the size of possible losses increases, each investor requires more than just 1-1/2 or 2 or 3 times the gain.

Graph 1 shows the conditions under which each might accept a wager like the one just described — along with the result of such an experiment.

The decision to invest is a lot like our example of the coin toss. The risk of an investment is typically measured by the variability of its return: how much above or (more important) below average does the return tend to fluctuate?[xii]

Suppose two investments yield an average return of 10%, but one yields exactly 10% every year, while the other ranges randomly from 0% to 20%. According to the “privatizers,” investors should see both investments as equivalent, because the average return is the same. But rational investors will obviously choose the first investment. The first investment offers a return of exactly 10%, but the second investment has two parts: a 10% average return, and a 50/50 chance of either gaining or losing 10% — a coin toss. Therefore, the second investment requires a “risk premium” — a higher return to compensate for the higher risk. Because the average return is the same for both, the “risk-adjusted” return, after subtracting the risk premium, is lower on the second investment than on the first.

The risk premium is the sum of the possible losses and the required offsetting gains, weighted by their probabilities[xiii]. On an investment with 10% average volatility, the typical investor requires a risk premium of about 1.8%, the conservative investor about 4.1%, and the speculative investor about 1.0%. Graph 2 shows the risk/return tradeoffs which would be equivalent to a 0% absolutely safe return for each of our sample investors. (For the convenience of future researchers, we attach a table of risk premia.)

Notice that the risk premium is different at each level of risk: not a single number for all investments, but a curve tracing all the tradeoffs between risk and return that the same investor would consider equivalent to a single risk-free rate of return (in this case, 0%). A 1% risk-adjusted return would trace a parallel curve, 1 percentage point higher, and so on.

Notice also that the risk premium rises faster than risk: an investment with 20% average variability is twice as risky as one with 10% variability, but for each of our investors, the required risk premium is about four times as high.

To compare the returns on investments involving different risks, then, we must subtract the appropriate risk premium from the average return on each investment. This gives us the “risk-adjusted” return: we can use it to compare apples with apples.

4. The “Frontier” of Possible Investments

The other key aspect of risk is that seeking higher returns generally requires accepting more risk. And on the whole, the risk rises faster than the return.

We can see this by comparing the average return of different investments with their average variability, measured by the “standard deviation” of returns.

For many years, Ibbotson Associates has tracked the average risk vs. return since the end of 1925, for Treasury securities, long-term corporate bonds, common stocks, and small-company stocks.[xiv] The results for the years 1926 to 1996 are shown in Graph 3.

Treasury bills had the lowest average variability (4.2%), but also the lowest inflation-adjusted return: 0.6%. Corporate bonds returned an average of 2.4% beyond inflation, but with a variability of 10.0%. Common stocks yielded 7.4% on average, but with 20.4% average variability in return. Small-cap stocks had the highest average real return (9.2%) but also the highest average volatility: 33.5%.

However, investors do not ordinarily invest all their wealth in a single asset. By mixing different assets, it is usually possible to find a combination with a higher return (at any level of risk), or a lower risk (for any level or return) than is possible by investing in a single asset.[xv] The line tracing the highest return possible at each level of risk is sometimes called the “efficient frontier.” No rational investor should accept a lower return for the risk, and no higher average return is available.

What effect does Social Security have on this range of investment choices?

As we noted earlier, for those already retired the real return on Social Security has averaged about 9%. But the rate of return started much higher, and has declined over time. For the very first retirees in the 1940s, the rate of return on payroll taxes approached 20 percent. For those retired from 1960 to 1968, the real rate of return averaged about 12%; from 1969 to 1975, about 9%; from 1976 to 1981, about 8%, and from 1982 to 1987, about 6%.[xvi] (Graph 4)

What we wish to focus on is what will happen when the average rate of return on Social Security levels out. The average rate of return of “steady-state” Social Security will be about the same as the growth rate of the economy. Since we wish to compare this return with the risks and real returns of financial investments from 1926 to 1996, we need to find out the average growth rate of the economy (adjusted by the same price index), as well as itsvolatility, during the same period. From 1926 to 1996, real GDP grew at an average rate of about 3.2%, with a standard deviation of about 5.5%.[xvii]

Graph 5 shows how the range of investment possibilities would have looked if pay-as-you-go Social Security had already reached its “steady state” before 1926. With “steady-state” Social Security, the return is higher at each level of risk, for all portfolio combinations between investing 100% in Treasury bills and 100% in common stocks.

However, one further adjustment is necessary. The return actually received by investors on financial assets or on Social Security is reduced by administrative and management fees. It is generally agreed that the relative size of administrative costs of Social Securty is vastly smaller than the fees charged on private financial accounts.[xviii] Put on the same basis, Social Security administrative costs amount to about 4 basis points (4 hundredths of a percentage point)[xix]; but the management fees on private financial accounts average close to 100 basis points (1 full percentage point).[xx]

Graph 6 shows how management fees affects the range of investment possibilities. The fees lower the return at each level of risk. But the reduction is slight to the extent that a portfolio includes Social Security, and relatively large to the extent that it includes stocks and bonds.

5. How Social Security Raises Investment Returns

We are now able to compare the risk-adjusted returns on “steady-state” Social Security and on financial investments.

The choice of investments actually made by each investor is the result of matching the range of possibilities with his or her attitude to risk. Each rational, well-informed investor will choose the portfolio that provides the highest risk-adjusted return.

Our “typical” investor also serves as a good representative of the median investor — one who is more risk averse than half the population, and less risk averse than the other half.

In the absence of Social Security, under the economic conditions from 1926 to 1996, the median investor would have realized the highest risk-adjusted return — 1.2% — with a portfolio consisting, on average, of about 55% Treasury bills and 45% common stocks.[xxi] But with steady-state Social Security, the median investor chooses a portfolio consisting of about 80% Social Security and 20% common stocks, which offers both a higher average return and lower average risk. The risk-adjusted return is 2.9% — more than twice as high. (Graph 7)

For our conservative investor, the gain due to Social Security is still larger. Without Social Security, the highest achievable real return, adjusted for risk, is zero — on a portfolio consisting of about 80% Treasury bills and 20% common stocks. Every portfolio with either a higher or lower average return has a negative risk-adjusted return. But Social Security raises the risk adjusted return by 2 percentage points. This return is possible with a portfolio consisting 80% of Social Security and 20% of common stocks. (Graph 8)

The biggest shift in portfolio (though not the largest increase in return) occurs with the speculative investor. Without Social Security, his highest achievable risk-adjusted return is 2.7%, on a portfolio consisting of about 75% common stocks and 25% intermediate Treasury securities. With Social Security, the speculative investor prefers a mix of about 60% Social Security and 40% common stocks. This shift causes the risk premium to fall much more than average return, so the risk-adjusted real return, net of fees, rises to 3.4%. (Graph 9)

This answers our original question. Does “steady-state” pay-as-you-go Social Security raise or lower the risk-adjusted real return on retirement saving for all future generations?

The answer, clearly, is that “steady-state” Social Security more than doubles the risk-adjusted real return for the median investor.[xxii] The evidence therefore argues against “privatizing” Social Security.

6. The “Privatizers” as Money Managers

But we are not quite finished. One of the most disturbing aspects of “privatization” schemes is that, although touted as an expansion of individual freedom, each would severely limit investment choices.

As one pair of proponents explains: “In fact, most proposals for a privatized national retirement system have regulatory elements that restrict investment strategies that are either too risky or that would be insufficiently aggressive to provide needed retirement benefits.”[xxiii]

Translation: In addition to abolishing pay-as-you-go retirement benefits, “privatization” plans would further reduce the risk-adjusted returns on retirement saving by imposing a one-size-fits-all approach to portfolio selection.

To see how this works, let’s consider the “model portfolios” put forward by the “privatizers.” Some are quite simple; others are enormously complex. But all have one thing in common: they do not allow for differences in risk aversion.

Peter Ferrara recommends a portfolio consisting 100% of common stocks, but argues that any mix of stocks and corporate bonds with an average yield higher than the rate of economic growth would be preferable to pay-as-you-go Social Security.[xxiv] Martin Feldstein also puts common stocks first, but falls back on a “conservative” portfolio, consisting of 60% common stocks and 40% long-term corporate bonds, or else 50% common stocks and 50% corporate bonds.[xxv] William Shipman has produced at least four model portfolios: two all-stock portfolios, one consisting of 75% large-company stocks and 25% small company stocks, the other 90% large-company stocks and 10% small company stocks; a “balanced” fund consisting of 54% common stocks, 6% small-company stocks, 20% long-term corporate bonds and 20% government bonds; and a long-term bond fund divided equally between corporate and government securities.[xxvi]

What is striking about all these choices is that each “privatizer” ranks the attractiveness of a portfolio purely by its 70-year average return, without regard to an individual investor’s attitude toward risk.

Let’s consider the effect that this cookie-cutter approach would have on risk-adjusted returns for our three kinds of individual investors. (For the moment, we will ignore a fact noted in a separate paper: for most families, a 20-year average return is appropriate. A 70-year average rate of return makes sense only for someone planning to retire at about age 165 — not age 65.)

We have already described the portfolios which are most efficient for each of our typical investors, with and without Social Security. So this time we will simply compare the real average risk-adjusted return on steady-state Social Security with the risk-adjusted returns of the model portfolios of the “privatizers,” and ignore the fact that more efficient portfolios are possible. The characteristics of each portfolio, as well as the risk-adjusted returns, are compared in a table.

Graph 10 shows the results for the typical or median investor. Steady-state Social Security alone beats every one of the “model” portfolio by a wide margin. What’s almost as interesting: the portfolios which the privatizers consider inferior outperform the portfolios which they consider superior. The portfolios consisting entirely of stocks actually have a negative return after adjusting for risk, while the “balanced funds” at least have a positive risk-adjusted yield.

Graph 11 shows the comparison for the conservative investor. This time, every one of the “privatizers'” model portfolios is sharply negative. Only steady-state Social Security offers a positive return.

Graph 12 compares the returns for the speculative investor. As might be expected, this case should offer the most favorable comparison for the model portfolios of the “privatizers”: for the speculative investor, at least, all the portfolios offer a positive return. Nevertheless, the risk-adjusted return of Social Security still beats the risk-adjusted return on each of the “model” portfolios.

7. Conclusion: Keep Social Security Pay-as-you-go.

Half a century of advances in portfolio theory and Wall Street practice seem to have been lost, as far as the “privatizers” are concerned. The “privatizers” labor under the impression that the way to improve investment returns is to assume more risk. The lesson that Wall Street has learned over the past few decades, on the contrary, is that the way to increase investment performance is by ruthlessly eliminating risk, either through diversification or by taking advantage of information which is not widely known.

By ignoring risk, the “privatizers” also ignore the most remarkable (and popular) characteristic of Social Security, considered as an investment: its “risk-adjusted” return is extraordinarily high, because its average volatility is quite low. The average future return on Social Security will indeed approximate the average rate of economic growth. And the variability of return on Social Security will also approximate that of the U.S. economy — which has been about one-quarter of the volatility of stock market returns.

After adjusting for this difference in risk, the average return on financial assets — or the stock market alone — has always been far lower than the average rate of economic growth. This means that the average risk-adjusted return on “steady-state” Social Security is higher than that of any class of financial investment. We have also shown that no possible combination of stocks and bonds could beat the risk-adjusted return of a portfolio that includes pay-as-you-go Social Security.

Finally, we have shown that the risk-adjusted returns of the “model portfolios” recommended by “privatizers” — who seek to make such portfolios mandatory — are inferior for nearly all investors. But the losses would be most acute for those investors who are most risk averse.

Based on the evidence, “privatizing” Social Security must lower, not raise, the total return on retirement saving.[xxvii]

Social Security Trustees (1997), The 1997 Annual Report of the Board of Trustees of the Federal Old-Age and Survivors Insurance and Disability Trust Funds, Ways and Means Committee, U.S. House of Representatives, April 24.

Thompson, Neil (1993), Portfolio Theory and the Demand for Money, St. Martin’s Press, New York.

Can Financial Assets Beat Social Security?Not in the Real World.a report by
John Mueller
Senior Vice President & Chief Economist
Lehrman Bell Mueller Cannon, Inc.for the National Committee to Preserve Social Security and Medicare
Washington, D.C.
October 1997

Summary

For those retired in the past 60 years, the average real return on Social Security payroll taxes — about 9% — has exceeded that of the stock market. But those who favor “privatizing” Social Security — replacing pay-as-you-go benefits with mandatory financial savings accounts — argue that future returns on financial assets will be higher than the return from Social Security.

Average future returns on Social Security, they point out, must approximate the average growth of the economy. Yet average real returns on financial assets since 1926 were higher — about 7% for a portfolio of common stocks, and about 5% for a mix of common stocks and corporate bonds, compared with 3% real economic growth.

However, the argument assumes that investors are indifferent to risk — the volatility of returns on investment. Because investors as a group are “risk-averse,” they do not seek the highest possible average return, but rather the highest risk-adjusted return. Just as nominal returns must be adjusted for inflation, real returns must be adjusted for risk.

The paper shows how to adjust returns on different investments for differences in risk. The average risk-adjusted returns from 1926 to 1996 on all classes of financial assets — including the stock market — were significantly lower than the rate of economic growth. So financial returns, under the same economic conditions, are lower than the average return on a “mature” pay-as-you-go Social Security system. The difference is still larger when the returns are measured net of management fees.

The paper also shows that the risk-adjusted return of a portfolio including Social Security can systematically exceed the return on a portfolio limited to financial assets. All of the model portfolios recommended by the “privatizers” — who seek to write them into law — fail to match the risk-adjusted return of “steady-state” Social Security.

Conclusion: ending pay-as-you-go Social Security must lower the total return on retirement saving.

[i] Duggan et al., 1993.

[ii] “Pay-as-you-go” means that workers’ current payroll taxes are used to pay for the current pensions of retired workers.

[iii] In a separate paper (“The Economics of Pay-as-you-go Social Security and the Economic Cost of Ending It”), we examine the history of economists’ thinking about Social Security, and compare it with economic experience. We conclude that this earlier view about Social Security, though it requires updating in some respects, was essentially correct in its conclusion.

[iv] Feldstein (1974), Feldstein (1977).

[v] Feldstein, 1997, 2-3.

[vi] The word “privatize” is used in quotation marks, because under all the “privatization” plans, the government would be called upon to closely regulate the kind of investments allowed. It is therefore debatable whether what is proposed is to “privatize” Social Security, or to “socialize” the private capital markets. Also, the economic arguments apply to replacing pay-as-you-go Social Security with any system of financial saving, public or private. Hence even some “privatizers,” like Martin Feldstein, prefer a different word.

[vii] Ferrara and Lott (1985), 32.

[viii] Ferrara and Lott (1985), 32.

[ix] This insight is usually attributed to Daniel Bernoulli (1738). An interesting but philosophically quirky history of risk theory can be found in Bernstein (1996). For a brief and readable but slightly more technical introduction, see “The Measurement of Utility and the Economics of Risk,” in McCloskey (1985), Chapter 2.

[x] A brief but comprehensive overview of the theory and research in this field can be found in MacCrimmon & Wehrung (1988), 44-50. MacCrimmon and Wehrung’s own study includes an experiment like the bet just described (shown for comparison in Graph 1). The authors describe flaws in the study’s design from the point of view of the original purpose (p. 120). Most conservative investors accepted the smallest bet and refused to gamble on the largest, so results of the smallest bet were skewed toward the most risk-averse investors and the largest bet was skewed toward the least risk-averse. For the purpose of the current paper this was fortunate, however, because it suggested three distinct subgroups with different attitudes toward risk, rather than just a single average.

[xi] The “utility function” described here is slightly different from two often used in theoretical discussions: the quadratic and the negative exponential. Both forms have two major drawbacks: they don’t seem to fit the observed facts, and neither is easily calculable by the investor whose decisions the function is supposed to describe. The negative exponential form makes people more risk-averse than they seem to behave, while the quadratic form can lead to absurd results (as explained in a readable appendix to Brealey, [1969], 133-139). The plain-vanilla utility function used in this paper has the advantages of fitting the facts — in controlled experiments as well as in observed investment returns — while being intuitively appealing and making the risk premium easy for an investor to calculate. Marginal utility declines inversely to wealth (k) raised to some power (s, s>0): U'(k) = k-s. Then total utility is U(k) = (1 – s)-1ak1-s + c, where a and c are scaling constants.

[xii] Volatility is typically measured by the standard deviation of returns. The standard deviation is measured in percent, but a 10% standard deviation with a 0% average return doesn’t exactly mean that the investment will be as likely to fall 10% the first year as to rise 10% the second. If that happened you would have only 99% of your original wealth. A 10% decline (to 9/10ths of your original wealth) the first year has to be followed by about an 11% gain (10/9ths) the second year to make up the loss. This is why investors learn to think, not in percent gains or losses of wealth, but in reciprocals of wealth.

Using the standard deviation as a measure of risk involves limiting assumptions which are too often ignored.

For example, it requires that fluctuations in the rate of return are individually random, but taken together have a probability that can be calculated: a so-called “normal” distribution. Therefore, using the standard deviation of returns as a measure of risk is valid only after we account for systematic factors — such as the growth of the economy, demography, and changing perceptions of risk — which can, at least in principle, be predicted.

Moreover, events which are individually unpredictable, but for which probability cannot be calculated, involve “uncertainty,” not risk. For example, the chance that Congress will unexpectedly change policy affecting Social Security benefits, or the taxation of investments in the stock market, cannot be stated as a matter of probabilities.

Finally, risk properly applies only to volatility involving below-average, not above-average returns; some have accordingly proposed different risk measures in cases where the positive and negative variance is not symmetrical.

Since we are not concerned in this paper to make a forecast, but only to determine what happened in the past — and because we will take these important qualifications into account in a separate paper — the standard deviation of returns can serve as a useful definition of risk.

Two justifiably popular surveys of research in this field are Malkiel (1996) and Johnson (1988).

[xiii] Our measurement of the risk premium amounts to the investor perceiving the return on an investment as including, in addition to some average return, a 50/50 “side bet” of either gaining or losing an amount equal to some multiple of the asset’s volatility, as measured by the (log) standard deviation of returns. The multiple corresponds to the investor’s aversion to risk: more risk-averse investors have a higher multiple. The “side bet” represents the largest (positive or negative) surprise likely to occur within some specified percentage of the time. If the events are in fact random, this simple choice yields a precise risk/return tradeoff and an easily calculated risk premium. For example, we saw that the typical investor seems to have a marginal utility function that varies with the inverse square of his wealth: with ratios of his wealth to the second power. Another way of interpreting this is to say that the typical investor appears to require about 95% certainty when dealing with random events — that is, allowing for events that might occur 1 year in 20. This is because about 95% of all random or “normal” variation in returns occurs within 2 standard deviations of the mean. If the extra standard deviation of an asset’s return is 10%, compared with some risk-free investment, then the required risk premium for this “typical” investor is .5(1.12) + .5/(1.12) – 1 = 1.82%.

Similarly, a conservative investor might allow for events that can be expected to occur at least 1 in 100 random events. About 99% of random events occur within 3 standard deviations of the mean, which corresponds to a marginal utility function that varies inversely with the cube of his wealth: ratios of wealth to the third power. So the risk premium for the conservative investor on an asset with a 10% standard deviation of returns is .5(1.13) + .5/(1.13) – 1 = 4.12%.

A relatively aggressive investor, our typical “commodity speculator,” might allow only for events that occur 1 in 4 times, which suggests a risk aversion factor of about 1.5. And so on.

[xiv] Ibbotson and Sinquefield (1997). The 1926 starting date is often chosen because the current Standard & Poor’s 500-stock index dates from that year — but the Cowles Commission index on which it is based goes back at least to 1880.

[xv] This “portfolio” return is determined not only by the average return on each investment, but also by the correlation of returns on different investments.

[xvi] Duggan et al., 1993, 10. Economists like Feldstein argue that these high rates of return on Social Security caused workers to save less, reducing total national investment. In a separate paper (“The Economics of Pay-as-you-go Social Security, and the Economic Cost of Ending It”), we show that Feldstein’s claim is based merely on re-defining investments in so-called “human capital” — expenses of child-rearing, education and training, health, safety and mobility of workers that increase earning ability — as “consumption.” In fact, the high initial rates of return on Social Security retirement saving appear to have played a major role in financing the Baby Boom — the most massive investment in “human capital” (so far) in world history.

[xvii] Only annual data for GDP are available until after World War II. Using annual data results in a slightly lower standard deviation than monthly data, upon which the Ibbotson calculations of standard deviation are based. However, this is offset by the fact that the volatility of wages, and therefore of the return on Social Security, is somewhat lower than for GDP. To calculate portfolio risks, it was also necessary to measure the correlation of GDP with each class of financial asset.

[xviii] For example, Hieger and Shipman (1997), 6.

[xix] Social Security administrative expenses last year comprised about $1.8 billion of total expenses of $308.2 billion: just under 0.6% (Social Security Trustees, 1997, 8). But from 1926 to 1996, about half the growth of nominal GDP was due to inflation; so administrative expenses would take about twice as large a share of the real rate of return on “steady-state” Social Security. 1.2% of the 3.2% growth rate of real GDP is about 0.04%, or 4 basis points.

[xx] 1994-96 Advisory Council on Social Security, 1997, Volume II, 487. The text notes that fees on IRA accounts tend to be higher than this. The figures cited, from Morningstar, Inc., were 99 basis points for equity funds, 84 basis points for balanced funds, 67 basis points for corporate bonds, and 89 basis points for government bonds. Our calculations will use these fees, and also assume 50 basis points for short-term securities.

[xxi] Such a portfolio had an average real return of 3.6% before fees, but 2.9% after fees. The portfolio’s 9.7% standard deviation requires a risk premium of 1.7%, leaving a risk-adjusted real return of 1.2%. The investor chooses this portfolio, because no portfolio with a higher average return compensates adequately for its additional risk; and no portfolio with lower risk offers a high enough average return.

[xxii] Since those who are risk averse, by all accounts, far outnumber the relative risk-takers, this is presumably true of the “average” investor as well as the median investor. However, the median investor is more appropriate than the average, because even a single investor with absolute aversion to risk would raise the average risk premium to infinity.

[xxiii] Hieger and Shipman (1997), 6.

[xxiv] Ferrara and Lott (1985).

[xxv] Feldstein (1977) and Feldstein and Samwick (1997).

[xxvi] Shipman (1995). Hieger and Shipman (1997).

[xxvii] In addition, phasing out pay-as-you-go Social Security would necessarily involve a large transition cost. The last generation covered would have to “pay twice for retirement” — its parents’ and its own — to finance the transition from pay-as-you-go to “pay-it-yourself” retirement pensions. The cost of ending pay-as-you-go Social Security is estimated in a separate paper.